On a reliable solution of a Volterra integral equation in a Hilbert space. (English) Zbl 1099.45001

Summary: We consider a class of Volterra-type integral equations in a Hilbert space. The operators of the equation considered appear as time-dependent functions with values in the space of linear continuous operators mapping the Hilbert space into its dual. We are looking for maximal values of cost functionals with respect to the admissible set of operators. The existence of a solution in the continuous and the discretized form is verified. The convergence analysis is performed. The results are applied to a quasistationary problem for an anisotropic viscoelastic body made of a long memory material.


45D05 Volterra integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
49J22 Optimal control problems with integral equations (existence) (MSC2000)
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[1] I. Bock, J. Lovíšek: Optimal control of a viscoelastic plate bending. Math. Nachr. 125 (1986), 135-151. · Zbl 0606.73104
[2] I. Bock, J. Lovíšek: An optimal control problem for a pseudoparabolic variational inequality. Appl. Math. 37 (1992), 62-80. · Zbl 0772.49008
[3] R. M. Christensen: Theory of Viscoelasticity. Academic Press, New York, 1982.
[4] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Applications 4, North Holland, Amsterdam, 1978. · Zbl 0383.65058
[5] J. Chleboun: On a reliable solution of a quasilinear elliptic equation with uncertain coefficients: sensitivity analysis and numerical examples. Nonlinear Anal. 44 (2001), 375-388. · Zbl 1002.35041
[6] I. Hlaváček: Reliable solution of linear parabolic problems with respect to uncertain coefficients. Z. Angew. Math. Mech. 79 (1999), 291-301. <a href=”http://dx.doi.org/10.1002/(SICI)1521-4001(199905)79:53.0.CO;2-N” target=”_blank”>DOI 10.1002/(SICI)1521-4001(199905)79:53.0.CO;2-N |
[7] I. Hlaváček: Reliable solution of problems in the deformation theory of plasticity with respect to uncertain material function. Appl. Math. 41 (1996), 447-466. · Zbl 0870.65095
[8] I. Hlaváček: Reliable solution of a torsion problem in Hencky plasticity with uncertain yield function. Math. Models Methods Appl. Sci. 11 (2001), 855-865. · Zbl 1037.74028
[9] I. Hlaváček: Reliable solution of a a perfect plastic problem with uncertain stress-strain law and yield function. SIAM J. Numer. Anal. 39 (2001), 1531-1555. · Zbl 1014.74015
[10] J. Kačur: Method of Rothe in Evolution Equations. Teubner, Leipzig, 1985. · Zbl 0582.65084
[11] J. Kačur: Application of Rothe’s method to integro-differential equations. J. Reine Angew. Math. 388 (1988), 73-105. · Zbl 0638.65098
[12] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Praha, 1967. · Zbl 1225.35003
[13] J. Nečas, I. Hlaváček: Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction. Studies in Applied Mathematics 3. Elsevier, 1981.
[14] K. Rektorys: The Method of Discretization in Time and Partial Differential Equations. Reidel, Dordrecht-Boston-London, 1982. · Zbl 0522.65059
[15] S. Shaw, J. R. Whiteman: Adaptive space-time finite element solution for Volterra equations arising in viscoelastic problems. J. Comput. Appl. Math. (4) 125 (2000), 1234-1257. · Zbl 0990.74071
[16] J. Simon: Compact sets in the space \(L^p(0,T;B)\). Ann. Mat. Pura Appl., IV. Ser. 146 (1987), 65-96. · Zbl 0629.46031
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